In principle there are three main methods for \(L_p\) -theory: multiplier theory, Calderón-Zygmund theory
and the pointwise estimate using sharp functions.
Multiplier theory fits well when the principal operator is almost
Laplacian and the equation under consideration is defined on the entire space, and Calderón-Zygmund theory works well when there exists an integral representation of solutions and the integral is taken over
\(\mathbb {R}^n\)  for some \(n\) . However, these two methods do not
fit our case since we are dealing with weighted \(L_p\) -theories for systems (REF ) and (REF ) defined on a half space. Thus we use an approach based on
pointwise estimates of the sharp function of second order derivatives, but unlike the standard
theory (for instance, [1]}) we need to use the weighted version.
The elaboration of this approach is one of our main results.
